An alternative method to crossing minimization on hierarchical graphs

A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of $k$ levels and then, as a second step, to permute the verti ces within the levels such that the number of crossings is minimized. We suggest an alternative method for the second step, namely, removing the minimal number of edges such that the resulting graph is $k$-level planar. For the final diagram the removed edges are reinserted into a $k$-level planar drawing. Hence, i nstead of considering the $k$-level crossing minimization problem, we suggest solv ing the $k$-level planarization problem. In this paper we address the case $k=2$. First, we give a motivation for our appro ach. Then, we address the problem of extracting a 2-level planar subgraph of maximum we ight in a given 2-level graph. This problem is NP-hard. Based on a characterizatio n of 2-level planar graphs, we give an integer linear programming formulation for the 2-level planarization problem. Moreover, we define and investigate the polytop e \2LPS(G) associated with the set of all 2-level planar subgraphs of a given 2 -level graph $G$. We will see that this polytope has full dimension and that the i nequalities occuring in the integer linear description are facet-defining for \2L PS(G). The inequalities in the integer linear programming formulation can be separated in polynomial time, hence they can be used efficiently in a branch-and-cut method fo r solving practical instances of the 2-level planarization problem. Furthermore, we derive new inequalities that substantially improve the quality of the obtained solution. We report on extensive computational results

Computational Molecular Biology

Computational Biology is a fairly new subject that arose in response to the computational problems posed by the analysis and the processing of biomolecular sequence and structure data. The field was initiated in the late 60's and early 70's largely by pioneers working in the life sciences. Physicists and mathematicians entered the field in the 70's and 80's, while Computer Science became involved with the new biological problems in the late 1980's. Computational problems have gained further importance in molecular biology through the various genome projects which produce enormous amounts of data. For this bibliography we focus on those areas of computational molecular biology that involve discrete algorithms or discrete optimization. We thus neglect several other areas of computational molecular biology, like most of the literature on the protein folding problem, as well as databases for molecular and genetic data, and genetic mapping algorithms. Due to the availability of review papers and a bibliography this bibliography

A note on computing a maximal planar subgraph using PQ-trees

The problem of computing a maximal planar subgraph of a non planar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQ-trees. The latest attempt has been reported by Jayakumar et al. [10]. In this paper we show that the algorithm presented by Jayakumar et al. is not correct. We show that it does not necessarily compute a maximal planar subgraph and we note that the same holds for a modified version of the algorithm presented by Kant [12]. Our conclusions most likely suggest not to use PQ-trees at all for this specific problem

Automatisiertes Zeichnen von Diagrammen

Dieser Artikel wurde für das Jahrbuch 1995 der Max-Planck-Gesellschaft geschrieben. Er beinhaltet eine allgemein verständliche Einführung in das automatisierte Zeichnen von Diagrammen sowie eine kurze Übersicht üuber die aktuellen Forschungsschwerpunkte am MPI

Real space first-principles derived semiempirical pseudopotentials applied to tunneling magnetoresistance

In this letter we present a real space density functional theory (DFT) localized basis set semi-empirical pseudopotential (SEP) approach. The method is applied to iron and magnesium oxide, where bulk SEP and local spin density approximation (LSDA) band structure calculations are shown to agree within approximately 0.1 eV. Subsequently we investigate the qualitative transferability of bulk derived SEPs to Fe/MgO/Fe tunnel junctions. We find that the SEP method is particularly well suited to address the tight binding transferability problem because the transferability error at the interface can be characterized not only in orbital space (via the interface local density of states) but also in real space (via the system potential). To achieve a quantitative parameterization, we introduce the notion of ghost semi-empirical pseudopotentials extracted from the first-principles calculated Fe/MgO bonding interface. Such interface corrections are shown to be particularly necessary for barrier widths in the range of 1 nm, where interface states on opposite sides of the barrier couple effectively and play a important role in the transmission characteristics. In general the results underscore the need for separate tight binding interface and bulk parameter sets when modeling conduction through thin heterojunctions on the nanoscale.Comment: Submitted to Journal of Applied Physic

Planar Embeddings with Small and Uniform Faces

Motivated by finding planar embeddings that lead to drawings with favorable aesthetics, we study the problems MINMAXFACE and UNIFORMFACES of embedding a given biconnected multi-graph such that the largest face is as small as possible and such that all faces have the same size, respectively. We prove a complexity dichotomy for MINMAXFACE and show that deciding whether the maximum is at most $k$ is polynomial-time solvable for $k \leq 4$ and NP-complete for $k \geq 5$. Further, we give a 6-approximation for minimizing the maximum face in a planar embedding. For UNIFORMFACES, we show that the problem is NP-complete for odd $k \geq 7$ and even $k \geq 10$. Moreover, we characterize the biconnected planar multi-graphs admitting 3- and 4-uniform embeddings (in a $k$-uniform embedding all faces have size $k$) and give an efficient algorithm for testing the existence of a 6-uniform embedding.Comment: 23 pages, 5 figures, extended version of 'Planar Embeddings with Small and Uniform Faces' (The 25th International Symposium on Algorithms and Computation, 2014

Interactive Exploration of Chemical Space with Scaffold Hunter

The supporting information is composed of the following files: I. pyruvatekinasedata.zip The pyruvate kinase data set used for the analysis described in the referenced publication is contained in this file. The analysis is based on the Pyruvate Kinase Screen as published in PubChem under the assay ID 361. It contains all compounds checked in this screen together with the scaffold tree generated from it. Scaffold Hunter can be used to query the database and interactively display the scaffold tree. This file is a dump from a MySQL 5.1 database and was generated with MySQL Administrator 1.2.5. It can be restored with the same program. II. scaffoldhunter_profiles.zip Scaffold Hunter saves the user profiles either on the hard disk or in a database. The corresponding database schema is contained in this zip file. This schema must be contained in the MySQL database before Scaffold Hunter can be run. This file is a dump from a MySQL 5.1 database and was generated with MySQL Administrator 1.2.5. It can be restored with the same program. III. InstallationGuide_Databases.pdf This document describes the installation of a local MySQL database server and the graphical user interface MySQL Administrator. Restoration of the profiles and sample databases are also described. IV. run_ScaffoldHunter.bat Windows batch file to run Scaffold Hunter with 1024 MByte of Memory. V. run_ScaffoldTreeGenerator.bat Windows batch file to run ScaffoldTreeGenerator with 1024 MByte of Memory. VI. ScaffoldHunter_readme.txt Textfile with advice for the installation of Scaffold Hunter. VII. ScaffoldTreeGenerator_readme.txt Textfile with advice for the installation of ScaffoldTree Generator

Graph Treewidth and Geometric Thickness Parameters

Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth $k$, the maximum thickness and the maximum geometric thickness both equal $\lceil{k/2}\rceil$. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth $k$, the maximum book thickness equals $k$ if $k \leq 2$ and equals $k+1$ if $k \geq 3$. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

Atomic matter wave scanner

We report on the experimental realization of an atom optical device, that allows scanning of an atomic beam. We used a time-modulated evanescent wave field above a glass surface to diffract a continuous beam of metastable Neon atoms at grazing incidence. The diffraction angles and efficiencies were controlled by the frequency and form of modulation, respectively. With an optimized shape, obtained from a numerical simulation, we were able to transfer more than 50% of the atoms into the first order beam, which we were able to move over a range of 8 mrad.Comment: 4 pages, 4 figure

Using Sifting for k-Layer Straightline Crossing Minimization

We present a new algorithm for k-layer straightline crossing minimization which is based on sifting that is a heuristic for dynamic reordering of decision diagrams used during logic synthesis and formal verification of logic circuits. The experiments prove sifting to be very efficient. In particular it outperforms the traditional layer by layer sweep based heuristics known from literature by far when applied to k-layered graphs with k \ge 3